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Vibrational properties of molecules are of widespread interest and importance in chemistry and biochemistry. The reliability of widely employed approximate computational methods is questioned here against the complex experimental spectrum of ethylene glycol. Comparisons between quantum vibrational self-consistent field and virtual-state configuration interaction (VSCF/VCI), adiabatically switched semiclassical initial value representation (AS SCIVR), and thermostatted ring polymer molecular dynamics (TRPMD) calculations are made using a full-dimensional machine-learned potential energy surface. Calculations are done for five low-lying conformers and compared with the experiment, with a focus on the high-frequency, OH-stretches, and CH-stretches, part of the spectrum. Fermi resonances are found in the analysis of VSCF/VCI eigenstates belonging to the CH-stretching band. Results of comparable accuracy, quality, and level of detail are obtained by means of AS SCIVR. The current VSCF/VCI and AS-SCIVR power spectra largely close the gaps between the experiment and TRPMD and classical MD calculations. Analysis of these results provide guidance on what level of accuracy to expect from TRPMD and classical MD calculations of the vibrational spectra for ubiquitous CH and OH-stretches bands. This work shows that even general vibrational features require a proper quantum treatment usually not achievable by the most popular theoretical approaches.

Most widely used machine learned (ML) potentials for condensed phase applications rely on many-body permutationally invariant polynomial (PIP) or atom-centered neural networks (NN). However, these approaches often lack chemical interpretability in atomistic energy decomposition and the computational efficiency of traditional force fields has not been fully achieved. Here, we present a novel method that combines aspects of both approaches, and achieves state-of-the-art balance of accuracy and force field-level speed. This method utilizes a monomer-centered representation, where the potential energy is decomposed into the sum of chemically meaningful monomeric energies. Without sophisticated neural network design, the structural descriptors of monomers are described by 1-body and 2-body effective interactions, enforced by appropriate sets of PIPs as inputs to the feed forward NN. We demonstrate the performance of this method through systematic assessments of models for gas-phase water trimer, liquid water, and also liquid CO2. The high accuracy, fast speed, and flexibility of this method provide a new route for constructing accurate ML potentials and enabling large-scale quantum and classical simulations for complex molecular systems.

Recent experiments have demonstrated that vibrational strong coupling (VSC) between molecular vibrations and the optical cavity field can modify vibrational energy transfer (VET) processes in molecular systems. However, the underlying mechanisms and the behavior of individual molecules under collective VSC remain largely incomplete. In this work, we combine state-of-the-art quantum vibrational spectral calculation, quantum wavepacket dynamics simulations, and ab initio machine-learning potential to elucidate how the vibrational dynamics of water OH stretches can be altered by VSC. Taking the (H$_2$O)$_{21}$-cavity system as an example, we show that the collective VSC breaks the localization picture, promotes the delocalization of OH stretches, and opens new intermolecular vibrational energy pathways involving both neighboring and remote water molecules. The manipulation of the VET process relies on the alignment of the transition dipole moment orientations of the corresponding vibrational states. The emergence of new energy transfer pathways is found to be attributed to cavity-induced vibrational resonance involving OH stretches across different water molecules, along with alterations in mode coupling patterns. Our fully quantum theoretical calculations not only confirm and extend previous findings on cavity-modified energy transfer processes but also provide new insights in energy transfer processes under collective VSC.

Progress in machine learning has facilitated the development of potentials that offer both the accuracy of first-principles techniques and vast increases in the speed of evaluation. Recently,"$\Delta$-machine learning" has been used to elevate the quality of a potential energy surface (PES) based on low-level, e.g., density functional theory (DFT) energies and gradients to close to the gold-standard coupled cluster level of accuracy. We have demonstrated the success of this approach for molecules, ranging in size from H$_3$O$^+$ to 15-atom acetyl-acetone and tropolone. These were all done using the B3LYP functional. Here we investigate the generality of this approach for the PBE, M06, M06-2X, and PBE0+MBD functionals, using ethanol as the example molecule. Linear regression with permutationally invariant polynomials is used to fit both low-level and correction PESs. These PESs are employed for standard RMSE analysis for training and test datasets, and then general fidelity tests such as energetics of stationary points, normal mode frequencies, and torsional potentials are examined. We achieve similar improvements in all cases. Interestingly, we obtained significant improvement over DFT gradients where coupled cluster gradients were not used to correct the low-level PES. Finally, we present some results for correcting a recent molecular mechanics force field for ethanol and comment on the possible generality of this approach.

Machine learning potentials (MLPs) are widely applied as an efficient alternative way to represent potential energy surfaces (PES) in many chemical simulations. The MLPs are often evaluated with the root-mean-square errors on the test set drawn from the same distribution as the training data. Here, we systematically investigate the relationship between such test errors and the simulation accuracy with MLPs on an example of a full-dimensional, global PES for the glycine amino acid. Our results show that the errors in the test set do not unambiguously reflect the MLP performance in different simulation tasks such as relative conformer energies, barriers, vibrational levels, and zero-point vibrational energies. We also offer an easily accessible solution for improving the MLP quality in a simulation-oriented manner, yielding the most precise relative conformer energies and barriers. This solution also passed the stringent test by the diffusion Monte Carlo simulations.

Here we assess two machine-learned potentials, one using the symmetric gradient domain machine learning (sGDML) method and one based on permutationally invariant polynomials (PIPs). These are successors to a PIP potential energy surface (PES) reported in 2004. We describe the details of both fitting methods and then compare the two PESs with respect to precision, properties, and speed of evaluation. While the precision of the potentials is similar, the PIP PES is much faster to evaluate for energies and energies plus gradient than the sGDML one. Diffusion Monte Carlo calculations of the ground vibrational state, using both potentials, produce similar large anharmonic downshift of the zero-point energy compared to the harmonic approximation the PIP and sGDML potentials. The computational time for these calculations using the sGDML PES is roughly 300 times greater than using the PIP one., Comment: 25 pages, 6 figures

Assessments of machine-learned (ML) potentials are an important aspect of the rapid development of this field. We recently reported an assessment of the linear-regression permutationally invariant polynomial (PIP) method for ethanol, using the widely used (revised) MD17 dataset. We demonstrated that the PIP approach outperformed numerous other methods, e.g., ANI, PhysNet, sGDML, p-KRR, with respect to precision and notably with respect to speed [Houston $et$ $al$., $J. Chem. Phys.$ 2022, 156, 044120.]. Here we extend this assessment to the 21-atom aspirin molecule, using the rMD17 dataset. Both energies and forces are used for training and the precision of several PIPs is examined for both. Normal mode frequencies, the methyl torsional potential, and 1d vibrational energies for an OH stretch are presented. Overall, we show that the PIPs approach outperforms other ML methods, including sGDML, ANI, GAP, PhysNet, and ACE, as reported by Kov\'acs $et$ $al.$ in $J. Chem. Theory$ $Comput.$ 2021, 17, 7696-7711.

The temperature dependence of the thermal rate constant for the reaction Cl($^2$P) + CH$_4$ $\rightarrow$ CH$_3$ + HCl is calculated using a Gaussian Process machine learning (ML) approach to train on and predict thermal rate constants over a large temperature range. Following procedures developed in two previous reports, we use a training dataset of approximately 40 reaction/potential surface combinations, each of which is used to calculate the corresponding data base of rate constant at approximately eight temperatures. For the current application, we train on the entire dataset and then predict the temperature dependence of the title reaction employing a "split" dataset for correction at low and high temperatures to capture both tunneling and recrossing. The results are an improvement on recent RPMD calculations compared to accurate quantum ones, using the same high-level ab initio potential energy surface. Both tunneling at low temperatures and recrossing at high temperatures are observed to influence the rate constants. Recrossing effects, which are not described by TST and even sophisticated tunneling corrections, do appear in experiment at temperatures above around 600 K. The ML results describe these effects and in fact, merge at 600 K with RPMD results (which can describe recrossing), and both are close to experiment at the highest experimental temperatures.

$\Delta$-Machine Learning ($\Delta$-ML) has been shown to effectively and efficiently bring a low-level ML potential energy surface to CCSD(T) quality. Here we propose extending this approach to general force fields, which implicitly or explicitly contain many-body effects. After describing this general approach, we illustrate it for the MB-pol water potential which contains CCSD(T) 2-body and 3-body interactions but relies on the TTM4-F 4-body and higher body interactions. The 4-body MB-pol (TTM4-F) interaction fails at a very short range and for the water hexamer errors up to 0.84 kcal/mol are seen for some isomers, owing mainly to 4-body errors. We apply $\Delta$-ML for the 4-body interaction, using a recent dataset of CCSD(T) 4-body energies that we used to develop a new water potential, q-AQUA. This 4-body correction is shown to improve the accuracy of the MB-pol potential for the relative energies of 8 isomers of the water hexamer as well as the harmonic frequencies. The new potential is robust in the very short range and so should be reliable for simulations at high pressure and/or high temperature.

Ethanol is a molecule of fundamental interest in combustion, astrochemistry, and condensed phase as a solvent. It is characterized by two methyl rotors and $trans$ ($anti$) and $gauche$ conformers, which are known to be very close in energy. Here we show that based on rigorous quantum calculations of the vibrational zero-point state, using a new ab initio potential energy surface (PES), the ground state resembles the $trans$ conformer but substantial delocalization to the $gauche$ conformer is present. This explains experimental issues about the identification and isolation of the two conformers. This "leak" effect is partially quenched when deuterating the OH group, which further demonstrates the need for a quantum mechanical approach. Diffusion Monte Carlo (DMC) and full-dimensional semiclassical dynamics calculations are employed. The new PES is obtained by means of a $\Delta$-Machine learning approach starting from a pre-existing low level (LL) density functional theory (DFT) surface. This surface is brought to the CCSD(T) level of theory using a relatively small number of $ab$ $initio$ CCSD(T) energies. Agreement between the corrected PES and direct $ab$ $initio$ results for standard fidelity tests is excellent. One- and two-dimensional discrete variable representation calculations focusing on the $trans$-$gauche$ torsional motion are also reported, in reasonable agreement with the experiment.

There has been great progress in developing methods for machine-learned potential energy surfaces. There have also been important assessments of these methods by comparing so-called learning curves on datasets of electronic energies and forces, notably the MD17 database. The dataset for each molecule in this database generally consists of tens of thousands of energies and forces obtained from DFT direct dynamics at 500 K. We contrast the datasets from this database for three "small" molecules, ethanol, malonaldehyde, and glycine, with datasets we have generated with specific targets for the PESs in mind: a rigorous calculation of the zero-point energy and wavefunction, the tunneling splitting in malonaldehyde and in the case of glycine a description of all eight low-lying conformers. We found that the MD17 datasets are too limited for these targets. We also examine recent datasets for several PESs that describe small-molecule but complex chemical reactions. Finally, we introduce a new database, "QM-22", which contains datasets of molecules ranging from 4 to 15 atoms that extend to high energies and a large span of configurations.

Many model potential energy surfaces (PESs) have been reported for water; however, none are strictly from "first principles". Here we report such a potential, based on a many-body representation at the CCSD(T) level of theory up to the ultimate 4-body interaction. The new PES is benchmarked for the isomers of the water hexamer for dissociation energies, harmonic frequencies, and unrestricted diffusion Monte Carlo (DMC) calculations of the zero-point energies of the Prism, Book, and Cage isomers. Dissociation energies of several isomers of the 20-mer agree well with recent benchmark energies. Exploratory DMC calculations on this cluster verify the robustness of the new PES for quantum simulations. The accuracy and speed of the new PES are demonstrated for standard condensed phase properties, i.e., the radial distribution function and the self-diffusion constant. Quantum effects are shown to be substantial for these observables and also needed to bring theory into an excellent agreement with experiment.

Permutationally invariant polynomial (PIP) regression has been used to obtain machine-learned (ML) potential energy surfaces, including analytical gradients, for many molecules and chemical reactions. Recently, the approach has been extended to moderate size molecules and applied to systems up to 15 atoms. The algorithm, including "purification of the basis", is computationally efficient for energies; however, we found that the recent extension to obtain analytical gradients, despite being a remarkable advance over previous methods, could be further improved. Here we report developments to compact further a purified basis and, more significantly, to use the reverse gradient approach to greatly speed up gradient evaluation. We demonstrate this for our recent 4-body water interaction potential. Comparisons of training and testing precision on the MD17 database of energies and gradients (forces) for ethanol against GP-SOAP, ANI, sGDML, PhysNet, pKREG, KRR, and other methods, which were recently assessed by Dral and co-workers, are given. The PIP fits are as precise as those using these methods, but the PIP computation time for energy and force evaluation is shown to be 10 to 1000 times faster. Finally, a new PIP PES is reported for ethanol based on a more extensive dataset of energies and gradients than in the MD17 database. Diffusion Monte Carlo calculations which fail on MD17-based PESs are successful using the new PES.

We report a permutationally invariant polynomial (PIP) potential energy surface for the water 4-body interaction. This 12-atom PES is a fit to 2119, symmetry-unique, CCSD(T)-F12a/haTZ (aug-cc-pVTZ basis for 'O' atom and cc-pVTZ basis for 'H' atom) 4-b interaction energies. These come from low-level, direct-dynamics calculations, tetramer fragments from an MD water simulation at 300 K, and from the water hexamer, heptamer, decamer, and 13-mer clusters. The PIP basis is purified to ensure that the 4-b potential goes rigorously to zero in monomer+trimer and dimer+dimer dissociations for all possible such fragments. The 4-b energies of isomers of the hexamer calculated with the new surface are shown to be in better agreement with benchmark CCSD(T) results than those from the MB-pol potential. Other tests validate the high-fidelity of the PES.

Machine-learned potential energy surfaces (PESs) for molecules with more than 10 atoms are typically forced to use lower-level electronic structure methods such as density functional theory and second-order Moller-Plesset perturbation theory (MP2). While these are efficient and realistic, they fall short of the accuracy of the ``gold standard'' coupled-cluster method, especially with respect to reaction and isomerization barriers. We report a major step forward in applying a $\Delta$-machine learning method to the challenging case of acetylacetone, whose MP2 barrier height for H-atom transfer is low by roughly 1.5 kcal/mol relative to the benchmark CCSD(T) barrier of 3.2 kcal/mol. From a database of 2151 local CCSD(T) energies, and training with as few as 430 energies, we obtain a new PES with a barrier of 3.49 kcal/mol in agreement with the LCCSD(T) one of 3.54 kcal/mol and close to the benchmark value. Tunneling splittings due to H-atom transfer are calculated using this new PES, providing improved estimates over previous ones obtained using an MP2-based PES.

``$\Delta$-machine learning" refers to a machine learning approach to bring a property such as a potential energy surface (PES) based on low-level (LL) density functional theory (DFT) energies and gradients to close to a coupled cluster (CC) level of accuracy. Here we present such an approach that uses the permutationally invariant polynomial (PIP) method to fit high-dimensional PESs. The approach is represented by a simple equation, in obvious notation $V_{LL{\rightarrow}CC}=V_{LL}+\Delta{V_{CC-LL}}$, and demonstrated for \ce{CH4}, \ce{H3O+}, and $trans$ and $cis$-$N$-methyl acetamide (NMA), \ce{CH3CONHCH3}. For these molecules, the LL PES, $V_{LL}$, is a PIP fit to DFT/B3LYP/6-31+G(d) energies and gradients, and $\Delta{V_{CC-LL}}$ is a precise PIP fit obtained using a low-order PIP basis set and based on a relatively small number of CCSD(T) energies. For \ce{CH4} these are new calculations adopting an aug-cc-pVDZ basis, for \ce{H3O+} previous CCSD(T)-F12/aug-cc-pVQZ energies are used, while for NMA new CCSD(T)-F12/aug-cc-pVDZ calculations are performed. With as few as 200 CCSD(T) energies, the new PESs are in excellent agreement with benchmark CCSD(T) results for the small molecules, and for 12-atom NMA training is done with 4696 CCSD(T) energies.